For the following simple graphs G1=(V1,E1) and G2=(V2,E2) (described by their vertex and edge sets) decide whether they are isomorphic. If they are then prove it, and if they are not give a convincing argument that explains why not. V1={a,b,c,d,e,f,g,h}, E1={ac,ag,ah,bc,bd,bh,ce,de,df,fg,fh}, V2={1,2,3,4,5,6,7,8}, E2={12,15,16,25,27,34,36,37,38,48,67}
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For the following simple graphs G1=(V1,E1) and G2=(V2,E2) (described by their vertex and edge sets) decide whether they are isomorphic.
If they are then prove it, and if they are not give a convincing argument that explains why not.
V1={a,b,c,d,e,f,g,h}, E1={ac,ag,ah,bc,bd,bh,ce,de,df,fg,fh},
V2={1,2,3,4,5,6,7,8}, E2={12,15,16,25,27,34,36,37,38,48,67}
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- For the following simple graphs G=(V,E) (described by their vertex and edge sets) decide whether they are bipartite or not. If G is bipartite, then give its bipartition, and if it is not explain why. 1) V={a,b,c,d,e,f,g}, E={ag,af,ae,bf,be,bc,dc,dg,df,de} 2) V={a,b,c,d,e,f,g,h}, E={ac,ad,ah,bc,bh,bd,ec,ef,eg,hf,hg}3. For the graph G= (V, E), find V, En all parallel edges, all loops and all isolated vertices and state whether G is a simple graph. Also state on which vertices edges es is incident and on which edges vertex v2 is incident. e1 U4Say that a graph G has a path of length three if there exist distinct vertices u, v, w, t with edges (u, v), (v, w), (w, t). Show that a graph G with 99 vertices and no path of length three has at most 99 edges.
- 3) The graph k-coloring problem is stated as follows: Given an undirected graph G= (V,E) with N vertices and M edges and an integer k. Assign to each vertex v in V a color c(v) such that 1Prove that the two graphs below are isomorphic.Two simple graphs are given with the following adjacency matrices. ro 1 1 11 ro 0 0 11 0 0 1 1 0 10 1| lí 1 1 0 Determine whether G and H are isomorphic to each other? Give reason(s). G = 1 10 0 li 0 0 o3) The graph k-coloring problem is stated as follows: Given an undirected graph G = (V,E) with N vertices and M edges and an integer k. Assign to each vertex v in Va color c(v) such that 1< c(v)Prove that any graph with the degree sequence 2,5, 5, 7, 7, 7,7, 7, 7 is Hamiltonian. A complete tripartite graph is formed by taking three groups of vertices A, B, and C, then adding an edge between every pair of vertices in different groups. We write Kabe for the complete tripartite graph with |A| = a, |B| = b, and |C| = c. For example, below are diagrams of K2,4,5 (left) and K23,6 (right). (a) One of these graphs is Hamiltonian. Find a Hamiltonian cycle in that graph. (b) The other of these graphs is not Hamiltonian. Give a reason why it does not have a Hamiltonian cycle. (c) Generalize: find and prove a rule that tells you when Kabe is Hamiltonian. You may assume aEvery set of vertices in a graph is biconnected if they are connected by two disjoint paths. In a connected graph, an articulation point is a vertex that would disconnect the graph if it (and its neighbouring edges) were removed. Demonstrate that any graph that lacks articulation points is biconnected. Given two vertices s and t and a path connecting them, use the knowledge that none of the vertices on the path are articulation points to create two disjoint paths connecting s and t.. Prove the following.(Note: Provide each an illustration for verification of results)Let H be a spanning subgraph of a graph G.i. If H is Eulerian, then G is Eulerian.ii. If H is Hamiltonian, then G is HamiltonianAn Euler path, in a graph or multigraph, is a walk through the graph which uses every edge exactly once. An Euler circuit is an Euler path which starts and stops at the same vertex. NOTE: graphs are in the image attached. Which of the graphs below have Euler paths? Which have Euler circuits? List the degrees of each vertex of the graphs above. Is there a connection between degrees and the existence of Euler paths and circuits? Is it possible for a graph with a degree 1 vertex to have an Euler circuit? If so, draw one. If not, explain why not. What about an Euler path? What if every vertex of the graph has degree 2. Is there an Euler path? An Euler circuit? Draw some graphs. Below is part of a graph. Even though you can only see some of the vertices, can you deduce whether the graph will have an Euler path or circuit? NOTE: graphs is in the image attached.Let G be a 4-regular plane graph with 10 faces. Determine how many vertices and edges G hasand give a plane drawing of such a graph. Please write complete sentences, include all details, show all of your work, and clarify all of your reasoning.SEE MORE QUESTIONSRecommended textbooks for youDatabase System ConceptsComputer ScienceISBN:9780078022159Author:Abraham Silberschatz Professor, Henry F. Korth, S. SudarshanPublisher:McGraw-Hill EducationStarting Out with Python (4th Edition)Computer ScienceISBN:9780134444321Author:Tony GaddisPublisher:PEARSONDigital Fundamentals (11th Edition)Computer ScienceISBN:9780132737968Author:Thomas L. FloydPublisher:PEARSONC How to Program (8th Edition)Computer ScienceISBN:9780133976892Author:Paul J. Deitel, Harvey DeitelPublisher:PEARSONDatabase Systems: Design, Implementation, & Manag…Computer ScienceISBN:9781337627900Author:Carlos Coronel, Steven MorrisPublisher:Cengage LearningProgrammable Logic ControllersComputer ScienceISBN:9780073373843Author:Frank D. PetruzellaPublisher:McGraw-Hill EducationDatabase System ConceptsComputer ScienceISBN:9780078022159Author:Abraham Silberschatz Professor, Henry F. Korth, S. SudarshanPublisher:McGraw-Hill EducationStarting Out with Python (4th Edition)Computer ScienceISBN:9780134444321Author:Tony GaddisPublisher:PEARSONDigital Fundamentals (11th Edition)Computer ScienceISBN:9780132737968Author:Thomas L. FloydPublisher:PEARSONC How to Program (8th Edition)Computer ScienceISBN:9780133976892Author:Paul J. Deitel, Harvey DeitelPublisher:PEARSONDatabase Systems: Design, Implementation, & Manag…Computer ScienceISBN:9781337627900Author:Carlos Coronel, Steven MorrisPublisher:Cengage LearningProgrammable Logic ControllersComputer ScienceISBN:9780073373843Author:Frank D. PetruzellaPublisher:McGraw-Hill Education